Packing fraction of geometric random packings of discretely
sized particles
Citation for published version (APA):
Brouwers, H. J. H. (2011). Packing fraction of geometric random packings of discretely sized particles. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 84(4), 042301-1/3. [042301].
https://doi.org/10.1103/PhysRevE.84.042301
DOI:
10.1103/PhysRevE.84.042301 Document status and date: Published: 01/01/2011
Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne Take down policy
If you believe that this document breaches copyright please contact us at: openaccess@tue.nl
providing details and we will investigate your claim.
PHYSICAL REVIEW E 84, 042301 (2011)
Packing fraction of geometric random packings of discretely sized particles
H. J. H. Brouwers*
Department of the Built Environment, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands
(Received 1 September 2011; published 24 October 2011)
The packing fraction of geometric random packings of discretely sized particles is addressed in the present paper. In an earlier paper [Brouwers,Phys. Rev. E 74, 031309 (2006);74, 069901(E) (2006)], analytical solutions were presented for the packing fraction of polydisperse geometric packings for discretely sized particles with infinitely large size ratio and the packing of continuously sized particles. Here the packing of discretely sized particles with finite size ratio u is analyzed and compared with empirical data concerning five ternary geometric random close packings of spheres with a size ratio of 2, yielding good agreement.
DOI:10.1103/PhysRevE.84.042301 PACS number(s): 45.70.Cc, 81.05.Rm
In Ref. [1] analytical solutions were presented for the packing fraction of polydisperse geometric packings. Two cases were distinguished, packings of discretely sized par-ticles with infinitely large size ratio u and the packing of continuously sized particles. It was shown that the latter dis-tribution follows from a discretely sized packing by taking the limit u→1.
The first case, u→∞, concerns noninteracting size groups, and the void fraction of the assembly can be derived from the packing fraction of the monosized packing of the particle shape considered (ϕ1) [2]. For a bimodal packing fraction, the minimum void fraction thus becomes ϕ2
1, in Ref. [3] designated as a saturated packing. In Fig.1 this principle is illustrated by the void fraction of packed bimodal broken particles with ϕ1= 0.5, taken from [3]. The figure reveals that the bimodal void fraction h is governed by the size ratio u and the composition of the mix. The composition can be expressed in the volume fraction of the large component (cL) or the
volume ratio of large and small components [r = cL/cS =
cL/(1–cL)], so h(u, r). For infinitely large size ratio, one can
see that the minimum bimodal packing fraction tends to 0.25, corresponding to ϕ2
1. The composition then corresponds to cL
= 2/3 (r = 2), corresponding to 1/(1 + ϕ1) [1].
Extending this situation of two noninteracting particle factions to n fractions yields ϕ1n as the saturated void fraction. For such saturated packing, i.e., a packing where the concentrations are such that a size class can fill the voids of the next larger size class, both the size ratio and the concentration ratio of subsequent particle classes is constant [1]. From the aforesaid void fraction expression one can see that the void fraction is reduced proportionally to the number of size groups minus 1, and is homogeneous on some long length scale. The bimodal void fraction, described by the function h(u, r), is also defined in the vicinity of u= 1. Considering that bimodal void fraction packing h(u, r) can range between ϕ1 and ϕ12, in Ref. [1] it was derived that the monosized void fraction ϕ1is reduced with a factor h/ϕ1when a second smaller fraction is added. When the size ratio u between the adjacent sizes in a multicomponent packing is finite, the perfect packing of smaller particles in the voids of the larger ones does not
*jos.brouwers@tue.nl
hold anymore, but also in this case the void fraction reduction involved with the size ratio of adjacent size groups (of constant ratio) is proportional to the number of size groups. Accordingly in Ref. [1], for a system with n size groups, was proposed
ϕ= ϕ1 h(u,r) ϕ1 n−1 . (1)
A maximum packing fraction is obtained when h is minimal. For a saturated packing of n sizes, Eq. (1) readily yields ϕn
1 indeed. The effect of adding an infinite number of size groups, to obtain a continuous packing, on the void fraction was also examined in Ref. [1]. Adding more size groups to the mix will reduce the void fraction. But on the other hand, its effect is less as the size ratio of adjacent groups tends to unity (i.e., u → 1) and the resulting void fraction of adjacent size groups, governed by h(u, r), tends to ϕ1. This limit was solved and the void fraction of a geometric continuous packing obtained. This expression was extensively validated by comparing it with a broad set of empirical data of [4], concerning continuous geometric packings of broken particles.
That Eq. (1) also holds for a finite number of interacting particles with finite size ratio is verified here. This is done by considering empirical ternary sphere packing results of Jeschar et al. [5], who did not specify how their packings were prepared and measured. They obtained ϕ1 = 0.366 for the monosized packing of spheres with diameters 7 mm (d3), 14 mm (d2), and 28 mm (d1). These values are compatible with random close packing (RCP) values from other experiments [6] and with computer generated values [7]. Equation (1) is, however, applicable to any mode of packing, from random loose to random close packed. To apply the current multimodal packing model, unimodal, bimodal, and trimodal packings only need to be prepared and compacted in a comparable way [1].
The bimodal and trimodal packing results from [5] are summarized in a ternary plot (Fig.2). The sides of the triangle show that the binary packings of 7 and 14 mm have the same void fraction as the mixes of 14 and 28 mm packing at equal compositions, which would be expected as the size ratios of these binary mixtures are equal (d1/d2 = d2/d3 = u = 2). These measured binary void fractions h are included in TableI for various compositions (cLor r).
042301-1
BRIEF REPORTS PHYSICAL REVIEW E 84, 042301 (2011)
ϕ
FIG. 1. Void fraction of bimodal mixes (h) of broken particles as a function of size ratio dL/dS (u) and volume fraction of large
constituent (cL) according to Furnas [3]. The ratio r [=cL/cS = cL/(1–cL)] is shown as well.
The compatible unimodal void fractions as measured with the three spheres sizes, as well as the compatible bimodal void fractions measured with d1/d2and d2/d3mixes, indicate that Jeschar et al. [5] conducted all their experiments in a reproducible and robust manner.
Using the void fraction of geometric ternary packings (n= 3) that can be extracted from Fig.2, it can now be verified if Eq. (1) is applicable. In Table I the compositions of five geometric packings are included, for which c1/c2= c2/c3 = r, r being 1/3, 1/2, 1, 2, and 3. The concentrations ci follow
from the cumulative finer fraction [1]: F(di)= dα i − d α 4 dα 1 − d α 4 , (2)
with i= 1, 2, 3, and 4, d4= 3.5 mm, and α from
α=ulog r, (3)
in which u is the base of the logarithm. The concentration ci of each size group follows from F (di)–F (di+1), and are
included in TableI. Equation (2) holds for α = 0; for α = 0 the cumulative finer function reads as follows:
F(d)=
u
log di−ulog d4
ulog d1−ulog d4. (4)
FIG. 2. Ternary void fractions of RCP of sphere mixes as a function of their composition according to Jeschar et al. [5].
Note that the packing consists of spheres with diameters 28, 14, and 7 mm, but that a size class of 3.5 mm is added (needed) in order to formulate the appropriate cumulative finer functions governed by Eqs. (2) and (4).
The ternary void fraction j (u, r) of each packing as measured by [5] is taken from Fig.2, and is included in TableI as well. Finally, the ternary void fraction is computed using Eq. (1), whereby the corresponding binary void fraction h(u, r) is taken from Fig.2and TableIas well. Comparing the values predicted by Eq. (1) and the measured values (see Table I) reveals that the ternary packing fraction can be predicted accurately from the binary void fraction h(u, r). So, in Ref. [1] it was seen that Eq. (1) holds for saturated discrete packings (u→ ∞) and continuous packings (u → 1); now it appears also to hold for geometric packings of discretely sized particles with finite size ratio (here: spheres with u= 2).
The author wishes to thank Dr. Dipl.-Ing. G. H¨usken for his assistance with the drawing of the figures.
TABLE I. Binary (h) and ternary (j ) void fractions as measured by Jeschar et al. [5] using discretely sized spheres (d1 = 28 mm, d2=
14 mm and d3= 7 mm, i.e., u = 2). Also the computed void fraction of a ternary geometric packing, using Eq. (1) with n= 3 and ϕ1= 0.366,
for various r (being ci/ci+1, or cL/cSin case of the binary mixture) is included.
cL cS r h(u, r) measureda α c1 c2 c3 j(u, r) measureda j(u, r) Eq. (1)
1/4 3/4 1/3 0.348 –1.58 0.08 0.23 0.69 0.334 0.333 1/3 2/3 1/2 0.344 –1 0.14 0.29 0.57 0.323 0.323 1/2 1/2 1 0.335 0 1/3 1/3 1/3 0.313 0.307 2/3 1/3 2 0.335 1 0.57 0.29 0.14 0.308 0.307 3/4 1/4 3 0.338 1.58 0.69 0.23 0.08 0.313 0.312 aReference [5]. 042301-2
BRIEF REPORTS PHYSICAL REVIEW E 84, 042301 (2011) [1] H. J. H. Brouwers, Phys. Rev. E 74, 031309 (2006); 74,
069901(E) (2006).
[2] C. C. Furnas,Ind. Eng. Chem. 23, 1052 (1931).
[3] C. C. Furnas, Department of Commerce, Bureau of Mines, Report of Investigation Serial No. 2894, 1928; Bulletin of US Bureau of Mines 307, 74 (1929).
[4] A. H. M. Andreasen and J. Andersen,Kolloid-Z. 50, 217 (1930).
[5] R. Jeschar, W. P¨otke, V. Petersen, and K. Polthier, in Proceedings
of the Symposium on Blast Furnace Aerodynamics, Wollongong,
Australia, 25th-27th September, 1975, edited by N. Standish
(Australasian Institute of Mining and Metallurgy, Illawara Branch, Wollongong, 1975), pp. 136–147.
[6] G. D. Scott, Nature (London). 188, 908 (1960); G. D. Scott and D. M. Kilgour, Br. J. Appl. Phys. (J. Phys. D) 2, 863 (1969).
[7] A. R. Kansal, S. Torquato, and F. H. Stillinger,
J. Chem. Phys. 117, 8212 (2002); Phys. Rev. E 66, 041109 (2002).
